TY - JOUR

T1 - Classification of some 3-subgroups of the finite groups of Lie type E6

AU - An, Jianbei

AU - Dietrich, Heiko

AU - Huang, Shih Chang

N1 - Funding Information:
Huang was supported by the Ministry of Science and Technology, Taiwan (project MOST 106-2115-M-006-012).

PY - 2018/12

Y1 - 2018/12

N2 - We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

AB - We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

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U2 - 10.1016/j.jpaa.2018.02.018

DO - 10.1016/j.jpaa.2018.02.018

M3 - Article

AN - SCOPUS:85042880063

VL - 222

SP - 4020

EP - 4039

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 12

ER -